Category Archives: Liberal arts math

I’m at a stopping point for the day in grading my GE 103 finals; the linear algebra final is coming up on Thursday. Unlike last year when I live-blogged my calculus final exam grading (it was on the old blog, no longer online), I’m not going to give a play-by-play of the grading here. Instead, I’ve been keeping a Sticky on my Mac’s desktop with a running list of thoughts as they come up. Here they are, relatively unpolished, with more probably to come as I grade more this week:

I’m not interested in teaching any subject or any course which does not specifically aim for, and does not specifically result in, the following: a significant change in the way students see the world, a significant increase in the creativity and curiosity of the students, and a significant refinement of the students’ ability to think analytically and learn on their own. Continue reading →

Tomorrow in GE 103 I had scheduled to cover an overview of continuous fair division problems. But I am grading the test they took this morning on voting and discrete fair division, and it is an absolute train wreck. The most basic principles and elementary math are being tortured to death. There is no way they are ready, willing, or able as a class to take on something new. So, what should I do? Should I:

Suck it up and teach the continuous fair division stuff anyway?

Work through each problem on the test “live” so they can see how it was supposed to be done?

So yesterday’s GE 103 lab assignment was basically just an extended group exercise on using the Adjusted Winner Procedure. The students were in pairs to simulate a business partnership that was splitting up. The problem said:

You need to divide up the following assets between the two of you: your business’ web site, the company car, the business’ computers, the business’ logo, and the office space you rented for your business’ operation. Use the Adjusted Winner Procedure to divide up these assets. Remember that you must begin by distributing 100 points among these five items…

First question: How many items are there to divide up? Hint: It SAYS SO EXPLICITLY IN THE LAST SENTENCE.

Guess what the most frequent mistake was on this problem? It wasn’t arithmetic or even algebra: it was the ability to recognize that there were five items being divided. Some teams put down four items to divide (they left off the office space). One team somehow invented a new item and put down six things. Never mind that the lab handout lists the five items and goes on to say that there are five items to distribute points across. How do you mess that up? It’s just inexplicable.

I have said it before, and will continue to say, that the biggest obstacle to learning mathematics is reading and writing. If you can’t even process basic verbal information in a problem, no amount of mathematical skill is going to help you solve it.

Here’s a cautionary tale for all those writing syllabi: In the syllabus for GE 103, I set different levels of collaboration allowed for different kinds of assignments. No collaboration on timed assessments; some collaboration allowed on the informal level for feedback journals but writeups to be done independently; and the following for labs:

I really, really, really should NOT have included lab homework in the category of “unlimited collaboration within your group”. It has led to groups in which one nice person does all the follow-up homework on a lab and the other two write down what the other person tells them to. I just had students turn in homework that was precisely identical, down to the punctuation marks and one student changing a word to match the other student’s word. And yet, by my policy, I can’t do anything about it because they’re in the same lab team. Big mistake on my part to allow that loophole.

Forget that rational choice theory clearly says that if you are going to copy someone’s homework and gain 6 points, but not understand the material and subsequently lose 20 points on the same question on a test, you cannot rationally choose to copy. I’m beginning to think that points alone are not a sufficient basis for applying that theory to the classroom; you have to look at the expense of points plus effort. An ill-advised student might choose to forfeit a few points — or more than a few — if it makes life less work-intensive.

Perhaps because the last two posts have cast a somewhat negative light on my GE 103 class, I wanted to share this entry from a Feedback Journal that a student in the class turned in last week. Each week I put a one-point “bonus” question on it that’s not really connected to the course material. This week, the question was: “What do you think about daylight savings time?” (Indiana has just switched back to DST after 30 years of not observing it.) This student’s answer?

I think daylight savings time gives people an excellent opportunity to experience the existential idea of time. Over spring break anyone who stayed in Indiana experienced a twenty-five hour day. This should throw everything off and it should be very hard to deal with since everything we do is built around the concept of time. Yet most people simply changed their clocks before they went to bed and woke up at 7:00 A.M. like they always do. An existentialist living in Indiana would have stayed awake until 12:00 and then set their clock back an hour and laughed at the thought that they were living the same hour over again. How joyful to realize how man-made time can seem in times like these. It really makes me wonder what it was like to live in Russia not a hundred years ago when they just jumped ahead a few days.

This is one of the reasons I enjoy teaching this class — you get people who think off the wall like this.

An entire lab period in GE 103 was spent on using the amortization formula to calculate the monthly payment on a house, given the price of the house, the annual interest rate, and the term of the mortgage. Students did four such calculations in class and two more individually for homework. Plus there were practice exercises assigned.

On the test, a question was asked using the identical format as on the lab, the homework, and the practice exercises. Out of 22 students:

Number of correct answers (with correct justifications): 7

Number of blank responses: 1

Number of responses that used the wrong formula (compound interest or savings formula being the most popular): 3

Number of responses using a version of amortization with a serious flaw (exponents turned into multiples, no “1 – ” term, etc.): 2

The other 9 responses set up the formula correctly but made some kind of procedural mistake, usually using the annual interest rate instead of the periodic interest rate, leading to monthly payments of somewhere in the range of $9000 per month on a $149,000 house with a 30-year mortgage.
Perhaps this is just having my standards set too high, but with as much practice as students have had on this kind of problem, and with this kind of problem being explicitly labelled on the review as being fair game, a 33% success rate is just unacceptable. How depressing — least of all for me. Like I said, there’s going to be a lot of unhappy people when I hand these tests back.

I’ve spent the better part of Friday and today grading GE 103 papers. In the process, I’m becoming more and more convinced that the single biggest roadblock to understanding mathematics, at least at the college level, is the ability to read and write. I don’t mean basic literacy; I mean the ability to consume and produce written information and use information effectively to solve problems. If a person can read and write effecitvely, s/he can make up whatever deficiencies may exist in their math background. If not, then not even the best prerequisite preparation is going to translate into success. Here are some cases in point:

A question on the latest GE 103 lab homework, which covers voting methods, gives a balloting situation and asks, “Use this example to show that the Hare system does not satisfy monotonicity.” Many of the answers consisted of: “The Hare system doesn’t satisfy monotonicity because [insert definition of monotonicity here].” Even if you don’t know the terms, it ought to be clear that just stating a definition proves nothing. You could ask me whether or not the sky is the color of an eggplant, and I could say, “Yes, because an eggplant is purple.” But it doesn’t work as an explanation, and of course is in fact a false statement. But for some reason many students don’t get that.

The latest GE 103 test covered probability. On the test is a clearly labelled instruction that says, “You must show all work and explain all your reasoning in order to receive credit for an answer. Answers that have no supporting work or explanations will receive 10% of the possible point value if they are correct and no credit at all if they are incorrect.” The bold face appears in the original on the test; this was also verbally announced at the beginning of the test. I have also repeatedly told the class that I am not grading their answer; I am grading the thought process which leads to the answer. Still, when asked to determine the probability of an event for which the answer is 1/2, well over the half the class just puts “1/2” with no justification. Is this is a failure of the ability to read (they glossed over the instructions) or the ability to write (they don’t realize that just writing a fraction doesn’t constitute a justification), or a linear combination of those? Some students lost tons and tons of credit on this test due to simply not justifying an answer after they were told to do so, even though justifying an answer is really easy and short and they were warned of the consequences of not doing so. It’s baffling. Plus, there are going to be a lot of unhappy people when they get their tests back and I am going to have to explain all over again why a disembodied answer isn’t enough.

I’m also seeing students make the same costly mistakes on tests that they made on homework, when I corrected the homework and told them what the mistake was. For example, one student lost points on a finance problem because s/he turned the exponent in the compound interest formula into a multple (A = P(1+i)n instead of A = P(1+i)n.) This is a nonzero but inexpensive error when caught on the homework level, and I made a clear comment on the homework about what the error was. 2 points lost. But now the same student has done the same thing on the test; 6 points lost. This is a reading problem; the student has the information to make the correction but then doesn’t covert it into actual correction. (Or maybe it’s a motivation problem; the student considers the effort spent on making and understanding the correction to be less valuable than the effort spent on something else.)

I single out GE 103 here, but of course this problem is pandemic. You get the same thing all the time in calculus, linear algebra, modern algebra, etc. (How many times have I asked, “Is the ring of integers mod 6 a field?” and gotten the answer “Yes, because every nonzero element is invertible”, ignoring the actual behavior of the ring?)

I’m strongly considering formatting future test and homework items in a two-tiered approach. Each problem would have two blanks, one labelled “Answer: ” and the other labelled “Justification: “. In a 10-point problem, the answer would be worth 1 point and the justification 9. That might succeed in getting across to students that the justification is what I am really grading here, and you can’t expect an answer to be self-evident.

And I will note that I can make this stuff as plain as day to students, but unless the individual student takes up the responsibility to act on the parameters that are set for them in their education, it won’t matter.